DOCSLIB.ORG

A Brief Introduction to Combinatorial Games

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Brief Introduction to Combinatorial Games A brief introduction to Combinatorial Games Asaf Ferber ∗ September 26, 2019 In these notes we give a brief introduction to combinatorial games. I used these notes few years ago for a mini-course that I gave for undergrads in MIT during the winter break (three lectures). Feel free to use/distribute it, and if you find errors/improve the presentation I would love to get an updated TEX file! (asaff@uci.edu). 1 Introduction How can one analyze a game? any attempt to analyze games like Chess, Go, Checkers, Tic-Tac- Toe, Hex etc. lead to the conclusion: there are enormously many cases to analyze and it seems quite hopeless. The only positive side is that it keeps the game interesting for a competition. As opposed to traditional Game Theory which focuses on incomplete information games (like Poker), here we consider perfect information games which are more similar in nature to Chess and Go. The term \Combinatorial Game" means a skill game, with no chance moves, where each player has a complete information about the current/past positions and can, theoretically, analyze all possible future position. The payoff function has 3 values: win, draw and loss. The first question one should ask: If these games are all deterministic, why can't we just use a computer to analyze? the simple answer is that even though in theory you can, practically there are just too many possibilities for any computer to analyze in a reasonable time. For example, consider a 3 dimensional version of the kids game Tic-Tac-Toe which is played on a board of size 5 × 5 × 5. This game has roughly 3125 positions (each position can be either marked \Player I", \Player II" or \Unoccupied"), which even though is only a finite number, it is so large that we would be afraid to meet it in a dark street at night. To summarize { traditional Game Theory is not useful for us, and a brute force computer search doesn't help even for very simple games as the 3 dimensions Tic-Tac-Toe mentioned above. What else can we do? 2 Examples and basic techniques In this section we discuss some examples and illustrate some useful techniques in analyzing certain games. ∗Department of Mathematics, UCI. Email: asaff@uci.edu. 1 2.1 Solitaire army The common feature of the Solitaire puzzles is that each one is played with a board and soldiers, the board contains a number of holes, each of which can hold 1 soldier. Each move consists of a jump by 1 soldier over 1 or more other soldiers. The soldiers jumped over being removed from the board. Each move therefore reduces the number of soldiers on the board. The Solitaire army is played on the infinite plane and the holes are in the lattice points Z × Z. The permitted moves are to jump horizontally or vertically. Suppose we start with all soldiers in the negative half plane. How many soldiers are needed to send one soldier forwards 1,2,3,4 or 5 holes into the upper plane? Obviously, 2 men are enough for sending one soldier 1 step forwards. Indeed, starting with the two soldiers standing on the same vertical line consecutively, the \lower" one can jump over the other and move 1 step forwards. It is also relatively easy to see that 4 soldiers are enough to send one 2 holes forwards (try it!). Less obvious is that eight soldiers are enough to push one soldier 3 holes forwards and 20 are enough to push a soldier 4 holes (exercise!). Before reading the next theorem, try to test your intuition by guessing the answer for the following question: Question 2.1. How many soldiers are needed to send one of them 5 holes forwards? To answer this question, probably the first thing I would do is to look at the online encyclopedia for integer sequences at https://oeis.org/. Based on that, the first search result which does not start at 1 gives us 52. It won't be very surprising if the answer is not 52 but some other, relatively small, number. But it is very surprising to know that: Theorem 2.2 (Conway, 1961). Impossible! Proof. The proof of this theorem pioneered the use of potential functions in combinatorics. The basic plan is quite simple: We wish to assign a weight to each hole with the condition that if ABC are three consecutive holes on a vertical/horizontal line with !(A);!(B);!(C) being their weights, then !(A) + !(B) ≥ !(C): (1) Suppose we have such a \weigh" function ! : Z × Z ! R, we evaluate a position in the game by the sum of all weights of holes which are occupied by soldiers { this sum is called the value of the position. Note that by 1 we have that each legal move cannot increase the value (prove it formally!). Therefore, such a weigh function proves that no matter how we paly, we can never get into a position of larger value from the initial position. The desired punch line will be to find such a function ! for which the position of having a soldier 5 steps forwards is larger than any initial position. 2 p Sounds easy? Let's do it! Let ! be a positive number satisfying ! + ! = 1 (the golden section { 5−1 2 ). Now, assume that we have managed to find a configuration of finitely many soldiers for which one soldier can move 5 holes forwards into the upper plane. Assign a weight 1 to the this hole in the positive upper plane, and extend it as follows: on the vertical line with the 1 on it downwards assign the weights 1; w; w2;:::. For each i ≥ 0, expand the horizontal line with !i to the left and right in a symmetric way by assigning the weights :::!i+`;!i+`−1;:::;!i+1; !i;!i+1;:::;!i+`−1;!i+`;:::. Now, a simple calculation gives us that the value of the top line of the lower half plane is w5 + 2w6 + 2w7 + ::: = w2: 2 P1 i 2 1 Therefore, the value of the whole lower half plane is i=2 ! = ! · 1−! = 1. In particular, since no hole in the lower plane has weight 0, no finite number of soldiers will suffice to send a man 5 forward! This completes the proof. 2.2 Strategy Stealing Strategy stealing is an existence argument which is oftenly used to determine the winner in a game between two perfect players. Unfortunately, it gives us no clue for how a winning/drawing strategy should look like. The strategy stealing argument is based on a symmetry argument, and therefore it works for symmetric games (whatever it means...). The class of games we are interested at in these notes are the so-called Positional Games. In a positional game there are two players, I and II, alternating turns in claiming previously unclaimed elements of some board V , with I going first. There is also a predetermined family of subsets of V , denoted by F which is considered as the collection of winning sets (the pair (V; F) is considered as the hypergraph of the game). The winner of the game is the first player to fully occupy all the elements of one of the winning sets in F. If there is no winner by the time that all the elements in V have been previously claimed, the game is declared as a draw. A complicated definition? not so... let us illustrate the definition by a simple example, namely the traditional child-game Tic-Tac-Toe. This game is played on a 3 × 3 board, where the players alternate turns in marking 0=X in previously unmarked spots. The winner is the first player to fully mark some combinatorial line (that is, horizontal, vertical, or diagonal line). This game can be easily described as a positional game! (do it). It sounds quite intuitive that the first player to move always have an advantage, and this is proven formally by the strategy stealing argument: Theorem 2.3 (Strategy stealing). Let (V; F) be any hypergraph of a positional game. Then, the first player can force at least a draw. Sketch of proof. Assume that the second player has a winning strategy and we wish to obtain a contradiction. The basic idea is to analyze what happens if the first player steals this strategy and plays accordingly. A winning strategy is a list of instructions telling the player how to answer each move. Now, Player I can play as follows: first move { arbitrary. From now on, she pretends to be the second player (and ignores her first move). If at some point she needs to claim the element she has claimed first, then she plays arbitrarily. The crucial observation here is that an extra move cannot harm any of the players. Let us now consider the following example: Consider a game with a board V = E(Kn) (that is, the board elements are all the edges of a complete graph on n vertices). Suppose that the winning sets consists of all subsets of edges that form a copy of Kk in Kn. We refer this game as the k-clique game played on E(Kn). A famous theorem of Ramsey asserts: Theorem 2.4 (Ramsey theorem). For every k there exists R(k) for which the following holds: every graph on at n ≥ R(k) many vertices either contains a copy of Kk or its complement (that is, the subgraph of Kn that consists of all the non-edges of G) contains a copy of Kk.
Load more

Recommended publications
  • Topics in Positional Games
    Tel Aviv University Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences Topics in Positional Games THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY by Asaf Ferber The research work for this thesis has been carried out under the supervision of Prof. Michael Krivelevich Submitted to the senate of Tel Aviv University July 2013 iii Acknowledgements It would not have been possible to write this dissertation without the help and support of the kind people around me, to only some of whom it is possible to give particular mention here. Above all, I would like to thank the one and only, my supervisor, Professor Michael Krivelevich, for his guidance, patience and continuous support throughout my studies. You are my role model of a great mathematician and a great teacher. It was a great honor to be your student. Special thanks to Danny Hefetz, who took me under his wings, acted as a second super- visor for me, and also turned into a close friend. I am very grateful for all you have done for me and I hope to continue this legacy with the younger generation. Moreover, I would like to thank all the professors from whom I had the pleasure and honor of learning. In particular, I would like to thank Noga Alon, Moti Gitik, Michael Krivelevich, Benny Sudakov and Tibor Szab´o. Next, I would like to thank all of my co-authors from all around the world. It was a great joy and honor to work with each of you! In particular I would like to thank my fellow graduate students from Berlin, Dennis Clemens and Anita Liebenau, for the continuous inspiring collaboration.
    [Show full text]
  • Globalwaiter-Client and Client-Waiter Games on Sparse Graphs
    ARIEL UNIVERSITY MASTER THESIS Global Waiter-Client and Client-Waiter Games on Sparse Graphs Author: Supervisor: Yael SABATO Prof. Dan HEFETZ Department of Computer Science December 25, 2018 i Declaration of Authorship I, Yael SABATO, hereby declare that this thesis proposal entitled, “Global Waiter- Client and Client-Waiter Games on Sparse Graphs” and the work presented in it are my own. I confirm that: • This work was done wholly or mainly while in candidature for a research de- gree at this University. • Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. • Where I have consulted the published work of others, this is always clearly attributed. • Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. • I have acknowledged all main sources of help. • Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed my- self. Signed: Date: ii Ariel University Abstract Faculty of Natural Sciences Department of Computer Science Master Global Waiter-Client and Client-Waiter Games on Sparse Graphs by Yael SABATO In this thesis we consider global Waiter-Client and Client-Waiter games, played on the edge set of sparse graphs. For a given global graph property P, we seek the smallest integer w^ =w ^(n; P) such that there is a graph G with v(G) = n and e(G) = w^, on which Waiter can force Client to build a subgraph that satisfies P in a Waiter- Client game on E(G), regardless of Client’s strategy.
    [Show full text]
  • A Threshold for the Maker-Breaker Clique Game∗
    A threshold for the Maker-Breaker clique game∗ Tobias M¨uller y MiloˇsStojakovi´c z October 7, 2012 Abstract We study the Maker-Breaker k-clique game played on the edge set of the random graph G(n; p). In this game, two players, Maker and Breaker, alternately claim unclaimed edges of G(n; p), until all the edges are claimed. Maker wins if he claims all the edges of a k-clique; Breaker wins otherwise. We determine that the threshold − 2 for the graph property that Maker can win this game is at n k+1 , for all k > 3, thus proving a conjecture from [20]. More precisely, we conclude that there exist − 2 constants c; C > 0 such that when p > Cn k+1 the game is Maker's win a.a.s., and − 2 when p < cn k+1 it is Breaker's win a.a.s. For the triangle game, when k = 3, we give a more precise result, describing the hitting time of Maker's win in the random graph process. We show that, with high probability, Maker can win the triangle game exactly at the time when a copy of K5 with one edge removed appears in the random graph process. As a consequence, we are able to give an expression for the limiting probability of Maker's win in the triangle game played on the edge set of G(n; p). 1 Introduction Let X be a finite set and let F ⊆ 2X be a family of subsets of X. In the positional game (X; F), two players take turns in claiming one previously unclaimed element of X.
    [Show full text]
  • Games on Graphs
    Games on graphs MiloˇsStojakovi´c? Department of Mathematics and Informatics, University of Novi Sad, Serbia [email protected] http://www.inf.ethz.ch/personal/smilos/ Abstract. Positional Games is a branch of Combinatorics which focuses on a variety of two player games, ranging from well-known games such as Tic-Tac-Toe and Hex, to purely abstract games played on graphs. The field has experienced quite a growth in recent years, with more than a few applications in related areas. We aim to introduce the basic notions, approaches and tools, as well as to survey the recent developments, open problems and promising research directions, keeping the main focus on the games played on graphs. Keywords: positional game, Maker-Breaker, Avoider-Enforcer, proba- bilistic intuition 1 Introduction Positional games are a class of combinatorial two-player games of perfect in- formation, with no chance moves and with players moving sequentially. These properties already distinguish this area of research from its popular relative, Game Theory, which has its roots in Economics. Some of the more prominent positional games include Tic-Tac-Toe, Hex, Bridg-It and the Shannon's switch- ing game. The basic structure of a positional game is fairly simple. Let X be a finite set and let F ⊆ 2X be a family of subsets of X. The set X is called the \board", and the members of F are referred to as the \winning sets". In the positional game (X; F), two players take turns in claiming previously unclaimed elements of X, until all the elements are claimed.
    [Show full text]
  • The Parameterized Complexity of Positional Games Edouard Bonnet, Serge Gaspers, Antonin Lambilliotte, Stefan Rümmele, Abdallah Saffidine
    The Parameterized Complexity of Positional Games Edouard Bonnet, Serge Gaspers, Antonin Lambilliotte, Stefan Rümmele, Abdallah Saffidine To cite this version: Edouard Bonnet, Serge Gaspers, Antonin Lambilliotte, Stefan Rümmele, Abdallah Saffidine. The Parameterized Complexity of Positional Games. ICALP, Jul 2017, Varsovie, Poland. 10.4230/LIPIcs. hal-01994361 HAL Id: hal-01994361 https://hal.archives-ouvertes.fr/hal-01994361 Submitted on 25 Jan 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Parameterized Complexity of Positional Games Édouard Bonnet1, Serge Gaspers2,3, Antonin Lambilliotte4, Stefan Rümmele5,2, and Abdallah Saffidine2 1 Middlesex University, London, UK [email protected] 2 The University of New South Wales, Sydney, Australia [email protected], [email protected] 3 Data61, CSIRO, Sydney, Australia 4 École Normale Supérieure de Lyon, Lyon, France [email protected] 5 The University of Sydney, Sydney, Australia [email protected] Abstract We study the parameterized complexity of several positional games. Our main result is that Short Generalized Hex is W[1]-complete parameterized by the number of moves. This solves an open problem from Downey and Fellows’ influential list of open problems from 1999.
    [Show full text]
  • The Critical Bias for the Hamiltonicity Game Is Ln 푛
    (1+표(1))푛 The Critical Bias for the Hamiltonicity Game is ln 푛 Aadil Shaikh, Rohit Chakravorty and Thai Ling April 2020 Abstract: This paper serves as an exposition on the theorem proved by Michael Krivelevich which states that the critical bias for the Hamiltonicity game between a Maker and Breaker (1+표(1))푛 can be generalised as [1]. Initially, basic introductions into some concepts used in ln 푛 Combinatorial Game Theory and Graph Theory are given. These are then built upon to create lemmas and theorems. Using these tools, the result is proved in multiple stages. 1. Introduction Popular two-person strategy games such as Tic-Tac-Toe (Noughts and Crosses) and Hex can be generalised as having a position where the players take turns making moves to achieve a defined winning condition. This gives way to the notion of the positional game used in Combinatorial Game Theory (CGT) which can be described by the following conditions: • 푋 - the board, with a finite set of elements known as positions • ℱ - the winning-sets, which are a family of subsets of 푋 • A criterion for winning the game Definition 1. A maker-breaker game is a type of positional game where two players, the Maker and the Breaker take it in turns to take unclaimed elements on a board. The objective of the Maker is to hold all the elements of a winning set, whereas the Breaker wins if they can prevent this (i.e. hold at least one element in each winning set). Alternatively, the Maker-Breaker game can be more formally defined as: a triple (퐻, 푎, 푏), with 퐻 = (푋, ℱ) where 퐻 is a hypergraph.
    [Show full text]
  • Problems in Positional Games and Extremal Combinatorics
    Problems in Positional Games and Extremal Combinatorics Christopher Kusch Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften am Fachbereich f¨urMathematik und Informatik der Freien Universit¨atBerlin Berlin, 2017 ii Erstgutachter: Prof. Tibor Szab´o,PhD (Betreuer) Zweitgutachter: Prof. Dr. Anusch Taraz Tag der Disputation: 4.10.2017 Contents Preface ix Acknowledgement xi 1 Introduction1 1.1 Strong Ramsey games...............................2 1.1.1 Introduction to strong games.......................2 1.1.2 The strong Ramsey game.........................3 1.1.3 The transition to the infinite board....................3 1.2 Maker-Breaker G-games and van der Waerden games..............5 1.2.1 Introduction to Maker-Breaker games..................5 1.2.2 The Maker-Breaker G-game........................7 1.2.3 The van der Waerden game and its generalisation............9 1.2.4 General winning criteria.......................... 11 1.3 Hypergraphs with Property O ........................... 13 1.3.1 Introduction of the problem........................ 13 1.3.2 Results: an improved upper bound.................... 15 1.4 Shattering extremal families............................ 15 1.4.1 Shattering.................................. 15 1.4.2 The elimination conjecture........................ 18 1.4.3 The results................................. 18 2 Strong Ramsey games: Drawing on an infinite board 21 2.1 Overview of the proof............................... 21 2.2 Sufficient conditions for a draw.......................... 22 2.2.1 No threat.................................. 24 2.2.2 Special threat................................ 25 2.2.3 Standard threats.............................. 27 iii iv CONTENTS 2.3 An explicit construction.............................. 29 2.4 Concluding remarks and open problems..................... 37 3 General Winning Criteria for Maker-Breaker games 39 3.1 The threshold bias of the k-AP game......................
    [Show full text]
  • Positional Games Involve Two Players Alternately Claiming Unoccupied Elements of a Set X, the Board of the Game; the Elements of X Are Called Vertices
    Positional Games Michael Krivelevich Abstract. Positional games are a branch of combinatorics, researching a variety of two- player games, ranging from popular recreational games such as Tic-Tac-Toe and Hex, to purely abstract games played on graphs and hypergraphs. It is closely connected to many other combinatorial disciplines such as Ramsey theory, extremal graph and set theory, probabilistic combinatorics, and to computer science. We survey the basic notions of the field, its approaches and tools, as well as numerous recent advances, standing open problems and promising research directions. Mathematics Subject Classification (2010). Primary 05C57, 91A46; Secondary 05C80, 05D05, 05D10, 05D40. Keywords. Positional games, Ramsey theory, extremal set theory, probabilistic intu- ition. 1. Introductory words Positional games are a combinatorial discipline, whose basic aim is to provide a mathematical foundation for analysis of two-player games, ranging from popular recreational games such as Tic-Tac-Toe and Hex to purely abstract games played on graphs and hypergraphs. Though the field has been in existence for several decades, motivated partly by its recreational side, it advanced tremendously in the last few years, maturing into one of the central branches of modern combinatorics. It has been enjoying mutual and fruitful interconnections with other combinatorial disciplines such as Ramsey theory, extremal graph and set theory, probabilistic arXiv:1404.2731v1 [math.CO] 10 Apr 2014 combinatorics, as well as theoretical computer science. The aim of this survey is two-fold. It is meant to provide a brief, yet gentle, introduction to the subject to those with genuine interest and basic knowledge in combinatorics. At the same time, we cover recent progress in the field, as well as its standing challenges and open problems.
    [Show full text]
  • On the Threshold for the Maker-Breaker H-Game
    ON THE THRESHOLD FOR THE MAKER-BREAKER H-GAME RAJKO NENADOV1, ANGELIKA STEGER1 AND MILOSˇ STOJAKOVIC´ 2 Abstract. We study the Maker-Breaker H-game played on the edge set of the random graph Gn;p. In this game two players, Maker and Breaker, alternately claim unclaimed edges of Gn;p, until all edges are claimed. Maker wins if he claims all edges of a copy of a fixed graph H; Breaker wins otherwise. In this paper we show that, with the exception of trees and triangles, the threshold for an H-game is given by the threshold of the corresponding Ramsey property of Gn;p with respect to the graph H. Keywords. Positional games; random graphs; Maker-Breaker 1. Introduction Combinatorial games are games like Tic-Tac-Toe or Chess in which each player has perfect information and players move sequentially. Outcomes of such games can thus, at least in principle, be predicted by enumerating all possible ways in which the game may evolve. But, of course, such complete enumerations usually exceed available computing powers, which keeps these games interesting to study. In this paper we take a look at a special class of combinatorial games, the so- called Maker-Breaker positional games. Given a finite set X and a family E of subsets of X, two players, Maker and Breaker, alternate in claiming unclaimed elements of X until all the elements are claimed. Unless explicitly stated otherwise, Maker starts the game. Maker wins if he claims all elements of a set from E, and Breaker wins otherwise. The set X is referred to as the board, and the elements of E as the winning sets.
    [Show full text]
  • Positional Games
    Mathematisches Forschungsinstitut Oberwolfach Report No. 44/2018 DOI: 10.4171/OWR/2018/44 Mini-Workshop: Positional Games Organized by Dan Hefetz, Ariel Michael Krivelevich, Tel Aviv Milos Stojakovic, Novi Sad Tibor Szabo, Berlin 30 September – 6 October 2018 Abstract. This mini-workshop focused on Positional Games and related fields. Positional Games Theory is a branch of Combinatorics whose main aim is to systematically develop an extensive mathematical basis for a variety of two-player games of perfect information and without chance moves, usually played on discrete objects. These include popular recreational games such as Tic-Tac-Toe and Hex as well as purely abstract games played on graphs and hypergraphs. Though a close relative of the classical Game Theory of von Neumann and of Nim-like games, popularized by Conway and others, Po- sitional Games are quite different and are more of a combinatorial nature. The subject is strongly related to several other branches of Combinatorics like Ramsey Theory, Extremal Graph and Set Theory, and the Probabilistic Method. It has also proven to be instrumental in deriving central results in Theoretical Computer Science, in particular in derandomization and algorith- mization of important probabilistic tools. Despite being a relatively young topic, there are already three textbooks dedicated to Positional Games as well as one invited talk at the International Congress of Mathematicians. Dur- ing this mini-workshop, several new exciting developments in the field were presented and discussed. We have also made some progress towards solving various open problems in Positional Games Theory and related areas. Mathematics Subject Classification (2010): 91A24 (Positional games), 05C57 (Games on graphs), 91A43 (Games involving graphs), 91A46 ()Combinatorial games), 05C65 (Hypergraphs), 05C80 (Random graphs), 05D40 (Probabilistic methods), 05D10 (Ramsey theory).
    [Show full text]
  • New Methods and Boards for Playing Tic-Tac-Toe
    University of Montana ScholarWorks at University of Montana Graduate Student Theses, Dissertations, & Professional Papers Graduate School 2012 Nontraditional Positional Games: New methods and boards for playing Tic-Tac-Toe Mary Jennifer Riegel The University of Montana Follow this and additional works at: https://scholarworks.umt.edu/etd Let us know how access to this document benefits ou.y Recommended Citation Riegel, Mary Jennifer, "Nontraditional Positional Games: New methods and boards for playing Tic-Tac- Toe" (2012). Graduate Student Theses, Dissertations, & Professional Papers. 707. https://scholarworks.umt.edu/etd/707 This Dissertation is brought to you for free and open access by the Graduate School at ScholarWorks at University of Montana. It has been accepted for inclusion in Graduate Student Theses, Dissertations, & Professional Papers by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. Nontraditional Positional Games: New methods and boards for playing Tic-Tac-Toe By Mary Jennifer Riegel B.A., Whitman College, Walla Walla, WA, 2006 M.A., The University of Montana, Missoula, MT, 2008 Dissertation presented in partial fulfillment of the requirements for the degree of Doctorate of Philosophy in Mathematics The University of Montana Missoula, MT May 2012 Approved by: Sandy Ross, Associate Dean of the Graduate School Graduate School Dr. Jennifer McNulty, Chair Mathematical Sciences Dr. Mark Kayll Mathematical Sciences Dr. George McRae Mathematical Sciences Dr. Nikolaus Vonessen Mathematical Sciences Dr. Michael Rosulek Computer Science Riegel, Mary J., Doctorate of Philosophy, May 2012 Mathematics Nontraditional Positional Games: New methods and boards for playing Tic-Tac-Toe Committee Chair: Jennifer McNulty, Ph.D.
    [Show full text]
  • A Threshold for the Maker-Breaker Clique Game∗
    A threshold for the Maker-Breaker clique game∗ Tobias M¨uller y MiloˇsStojakovi´c z Abstract We study the Maker-Breaker k-clique game played on the edge set of the random graph G(n; p). In this game, two players, Maker and Breaker, alternately claim un- claimed edges of G(n; p), until all the edges are claimed. Maker wins if he claims all the edges of a k-clique; Breaker wins otherwise. We determine that the threshold for the − 2 graph property that Maker can win this game is at n k+1 , for all k > 3, thus proving a conjecture from [10]. More precisely, we conclude that there exist constants c; C > 0 − 2 − 2 such that when p > Cn k+1 the game is Maker's win a.a.s., and when p < cn k+1 it is Breaker's win a.a.s. For the triangle game, when k = 3, we give a more precise result, describing the hitting time of Maker's win in the random graph process. We show that, with high probability, Maker can win the triangle game exactly at the time when a copy of K5 with one edge removed appears in the random graph process. As a consequence, we are able to give an expression for the limiting probability of Maker's win in the triangle game played on the edge set of G(n; p). 1 Introduction Let X be a finite set and let F ⊆ 2X be a family of subsets of X. In the positional game (X; F), two players take turns in claiming one previously unclaimed element of X.
    [Show full text]